In general, nuclear power systems don’t actually produce much waste material.
They periodically get their fuel swapped out, but that’s a good 20 – 30 years of service-life before that needs to happen.
Most nuclear waste is actually just irradiated material, things like hazmat suits, tools used to handle radioactive materials, misc debris.
Trash really. Very little is actual plutonium/uranium or anything like that.
Beyond that, a nuclear submarine’s reactor functions very much the same as a larger land-based reactor.
Radioactive material gets hotter the closer you pack it with more of itself, and you can use that heat to boil water into steam and drive turbines to generate electricity.
Basically some kinds of atom are ridiculously overcomplicated and will shed bits of themselves steadily until they reach a stable state. This can take thousands of years to finish.
These offcast neutrons don’t just disappear though, they fly off with some force and smack into other atoms and break them up in turn, producing more neutrons which smack into more atoms etc etc.
The tighter-packed the radioactive material, the faster this happens.
All of this breaking of atoms produces a lot of extra energy which manifests as heat.
So if you get a big pile of uranium, it will heat up. Do it right, it can boil water, which produces steam, which drives turbines, generates electricity and powers your submarine.
As a side-note:
The basic operation of a nuclear bomb is to pack the radioactive material as close together as possible and make the atoms pop as quickly as possible for maximum heat.
This is generally accomplished by use of conventional explosives to squeeze the uranium together, plus various extra stuff like Berylium lenses to produce additional neutrons to get a more thorough atom-popping.
Do it all at once and you get a lot of energy, and that’s gotta go somewhere. Generally it goes into a very large explosion.
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